Definition:Compact Linear Transformation/Normed Vector Space/Definition 1

Definition

Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.

Let $T : X \to Y$ be a linear transformation.

Let $\operatorname {ball} X$ be the closed unit ball in $\struct {X, \norm \cdot_X}$.

We say that $T$ is a compact linear transformation if and only if:

$\map \cl {\map T {\operatorname {ball} X} }$ is compact in $\struct {Y, \norm \cdot_Y}$

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where $\cl$ denotes topological closure.

Also see

• Equivalence of Definitions of Compact Linear Transformation on Normed Vector Space

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.4.1$